$N$-boson spectrum from a discrete scale invariance

We present an analysis of the $N$-boson spectrum computed using a soft two-body potential, the strength of which has been varied in order to cover an extended range of positive and negative values of the two-body scattering length $a$ close to the unitary limit. The spectrum shows a tree structure of two states, one shallow and one deep, attached to the ground state of the system with one less particle. It is governed by a unique universal function $\Delta \left(\xi \right)$, already known in the case of three bosons. In the three-particle system the angle $\xi$, determined by the ratio of the two- and three-body binding energies $E_3/E_2={tan}^2\xi$, characterizes the discrete scale invariance of the system. Extending the definition of the angle to the $N$-body system as $E_N/E_2={tan}^2\xi$, we study the $N$-boson spectrum in terms of this variable. The analysis of the results, obtained for up to $N=16$ bosons, allows us to extract a general formula for the energy levels of the system close to the unitary limit. Interestingly, a linear dependence of the universal function as a function of $N$ is observed at fixed values of $a$. We show that the finite-range nature of the calculations results in the range corrections that generate a shift of the linear relation between the scattering length $a$ and a particular form of the universal function. We also comment on the limits of applicability of the universal relations.

Figure 1

Universal function $y\left(\xi \right)$ as a function of $a_B$ in units of $r_0$. Numerical results are given by circles ($n=0$) and squares ($n=1$) with the dashed lines representing the best linear fits to the results (upper panel). The bottom panel shows a zoom close to the thresholds $y(-\pi )=-1.56$ and $y(-\pi /4)=0.071$ (given by the thick horizontal lines).

Figure 2

Universal function $y\left(\xi \right)$ as a function of $a_B$, in units of $r_0$ for $N=4–16$ for (a) ground-state energies and (b) excited-state energies. A zoom of the plots close to the $-1.56$ threshold is given for (c) $n=0$ and (d) $n=1$.

Figure 3

Ratio ${\kappa }_N^m/{\kappa }_3^0$ (top) and shift ${\Gamma }_N^m$ (bottom) as a function of $N$. The circles correspond to $m=0$ and the squares to $m=1$. The solid lines are the best linear fits.

Figure 4

Shift ${\Gamma }_N^m$ as a function of the ratio ${\kappa }_N^m/{\kappa }_3^0$ for $m=0$ (circles) and $m=1$ (squares).

Figure 5

Universal function $y\left(\xi \right)$ as a function of $a_B$, in units of $\ell$ for the ground state (solid lines) and excited state (dashed lines) close to the $y(-\pi )=-1.56$ threshold, for $N=4–8$. Starting with $N=8$, the excited state becomes unbound at this threshold as it crosses the ground-state line before arriving at the threshold.

Figure 6

Ratios ${\kappa }_{N+1}^0/{\kappa }_N^0$ (circles), ${\kappa }_{N+1}^1/{\kappa }_N^0$ (triangles), and ${\kappa }_{N+1}^1/{\kappa }_N^1$ (squares) as a function of $N$.

Figure 7

Dimensionless quantity ${\kappa }_N^n{a}_N^{n,-}$ as a function of $N$ for $n=0$ (top) and $n=1$ (bottom).

Figure 8

Universal function $y\left(\xi \right)$ as a function of $a_B$ calculated from the data of Ref. [34]. The solid line represents the best linear fit to the data.

Figure 9

Analysis of the results from Ref. [16] in the $(a,y)$ plane (top). In the bottom panel the ratios ${\kappa }_N^0/{\kappa }_3^0$ are given as a function of $N$ for the results of Ref. [16] (diamonds), the present results (circles), and the results from Ref. [35] (squares). The universal prediction is given by the dashed line. The solid and dash-dotted lines are fits to the data. The dotted line is a fit to the results of Ref. [16] up to $N=8$.

Figure 10

Universal function $y\left(\xi \right)$ as a function of $N$ calculated using the results of Ref. [36] for the TTY potential (circles), the results of Ref. [37] for the HFDHE2 potential (asterisks), the present results for the TBG interaction (squares), and the results of Ref. [17] using a two-body plus a three-body force (triangles). The straight lines are the best fit to the data.